ACCESSIBILITY AND ABUNDANCE OF ERGODICITY IN DIMENSION THREE: A SURVEY

ACCESSIBILITY AND ABUNDANCE OF ERGODICITY IN

DIMENSION THREE: A SURVEY.

F.RODRIGUEZ HERTZ, M.RODRIGUEZ HERTZ, AND R. URES

Abstract. In [18] the authors proved the Pugh-Shub conjecture for partiallyhyperbolic diffeomorphisms with 1-dimensional center, i.e. stably ergodic dif-feomorphisms are dense among the partially hyperbolic ones and, in subsequentresults [20, 21], they obtained a more accurate description of this abundance ofergodicity in dimension three. This work is a survey type paper of this subject.

1. Introduction

The purpose of this survey is to present the state of the art in the studyof the ergodicity of conservative partially hyperbolic diffeomorphisms in threedimensional manifolds. In fact, we shall mainly describe the results contained in[20, 21, 11]. The study of partial hyperbolicity has been one of the most activetopics on dynamics over the last years and we do not pretend to describe all therelated results, even for 3-manifolds. Some of the important themes excludedin this survey are entropy maximizing measures, absolute continuity of centerfoliations, co-cycles over partially hyperbolic systems, SRB-measures, dynamicalcoherence, classification, etc.

A diffeomorphism f : M → M of a closed smooth manifold M is partiallyhyperbolic if TM splits into three invariant bundles such that one of them iscontracting, the other is expanding, and the third, called the center bundle, hasan intermediate behavior, that is, not as contracting as the first, nor as expandingas the second (see the Subsection 2.3 for a precise definition). The first and secondbundles are called strong bundles.

A central point in dynamics is to find conditions that guarantee ergodicity.In 1994, the pioneer work of Grayson, Pugh and Shub [9] suggested that partialhyperbolicity could be “essentially” a sufficient condition for ergodicity. Indeed,soon afterwards, Pugh and Shub conjectured that stable ergodicity (open setsof ergodic diffeomorphisms) is dense among partially hyperbolic systems. Theyproposed as an important tool the accessibility property (see also the previouswork by Brin and Pesin [2]): f is accessible if any two points of M can bejoined by a curve that is a finite union of arcs tangent to the strong bundles.

Date: August 13, 2010.2000 Mathematics Subject Classification. Primary: 37D30, Secondary: 37A25.Key words and phrases. partial hyperbolicity; accessibility property; ergodicity; laminations.

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2 F.RODRIGUEZ HERTZ, M.RODRIGUEZ HERTZ, AND R. URES

Essential accessibility is the weaker property that any two measurable sets ofpositive measure can be joined by such a curve. In fact, accessibility will play akey role in this survey.

Pugh and Shub split their Conjecture into two sub-conjectures: (1) essentialaccessibility implies ergodicity, (2) the set of partially hyperbolic diffeomorphismscontains an open and dense set of accessible diffeomorphisms.

Many advances have been made since then in the ergodic theory of partiallyhyperbolic diffeomorphisms. In particular, there is a result by Burns and Wilkin-son [4] proving that essential accessibility plus a bunching condition (triviallysatisfied if the center bundle is one dimensional) implies ergodicity. There is alsoa result by the authors [18] obtaining the complete Pugh-Shub conjecture forone-dimensional center bundle. See [19] for a survey on the subject.

We have therefore that almost all partially hyperbolic diffeomorphisms withone dimensional bundle are ergodic. This means that the non-ergodic partiallyhyperbolic systems are very few. Can we describe them? Concretely,

Question 1.1. Which manifolds support a non-ergodic partially hyperbolic dif-feomorphism? How do they look like?

In this survey we give a description of what is known about this question forthree dimensional manifolds. We study the sets of points that can be joined bypaths everywhere tangent to the strong bundles (accessibility classes), and arrive,using tools of geometry of laminations and the topology of 3-manifolds, to thesomewhat surprising conclusion that there are strong obstructions to the non-ergodicity of a partially hyperbolic diffeomorphism. See Theorems 1.4, 1.7 and1.8.

This gave us enough evidence to conjecture the following:

Conjecture 1.2 ([20]). The only orientable manifolds supporting non-ergodicpartially hyperbolic diffeomorphisms in dimension 3 are the mapping tori of dif-feomorphisms of surfaces which commute with Anosov diffeomorphisms.

Specifically, they are (1) the mapping tori of Anosov diffeomorphisms of T2,(2) T3, and (3) the mapping torus of −id where id : T2 →T2 is the identity mapon the 2-torus.

Indeed, we believe that for 3-manifolds, all partially hyperbolic diffeomorphismsare ergodic, unless the manifold is one of the listed above.

In the case that M = T3 we can be more specific and we also conjecture that:

Conjecture 1.3. Let f : T3 → T3 be a conservative partially hyperbolic diffeo-morphism homotopic to a hyperbolic automorphism. . Then, f is ergodic.

In [20] we proved Conjecture 1.2 when the fundamental group of the manifoldis nilpotent:

ACCESSIBILITY AND ABUNDANCE OF ERGODICITY IN DIMENSION THREE 3

Theorem 1.4. All the conservative C2 partially hyperbolic diffeomorphisms ofa compact orientable 3-manifold with nilpotent fundamental group are ergodic,unless the manifold is T3.

A paradigmatic example is the following. Let M be the mapping torus of

Ak : T2 →T2, where Ak is the automorphism given by the matrix

(

1 k

0 1

)

, k

a non-zero integer. That is, M is the quotient of T2 × [0, 1] by the relation ∼,where (x, 1) ∼ (Akx, 0). The manifold M has nilpotent fundamental group; infact, it is a nilmanifold. Theorem 1.4 then implies that all conservative partiallyhyperbolic diffeomorphisms of M are ergodic.

Let us see that the above case, namely the case of nilmanifolds, is the onlyone where Theorem 1.4 is non-void. The Geometrization Conjecture, gives, afterPerelman’s work:

Theorem 1.5. If M is a compact orientable manifold with nilpotent fundamentalgroup, then either M is a nilmanifold or else it is finitely covered by S3 or S2×S1.

The second case mentioned in Theorem 1.5 is ruled out by a remarkable resultby Burago and Ivanov:

Theorem 1.6 ([3]). There are no partially hyperbolic diffeomorphisms in S3 orS2 × S1.

The proofs of most of the theorems of this survey involve deep results of thegeometry of codimension one foliations and the topology of 3-manifolds. In Sec-tion 2.1 we shall include, for completeness, the basic facts and definitions that weshall be using in this work. However, the interested reader is strongly encouragedto consult [5], [6], [12] and [13] for a well organized and complete introduction tothe subject.

Let us explain a little bit our strategy. In the first place, it follows from theresults in [4, 18] that accessibility implies ergodicity. So, our strategy will beto prove that all partially hyperbolic diffeomorphisms of compact 3-manifoldsexcept the ones of the manifolds listed in Conjecture 1.2 satisfy the (essential)accessibility property.

In dimension 3, and in fact, whenever the center bundle is 1-dimensional, thenon-open accessibility classes are codimension one immersed manifolds [18]; theset of non-open accessibility classes is a compact set laminated by the accessibilityclasses (see Section 2.1 for definitions). So, either f has the accessibility propertyor else there is a non-trivial lamination formed by non-open accessibility classes.

Let us first assume that the lamination is not a foliation (i.e. does not coverthe whole manifold). Then in [20] it is showed that it either extends to a truefoliation without compact leaves, or else it contains a leaf that is a periodic 2-toruswith Anosov dynamics. In the first case, we have that the boundary leaves of the


lamination contain a dense set of periodic points [18], and that their fundamentalgroup injects in the fundamental group of the manifold. In the second case, letus call any embedded 2-torus admitting an Anosov dynamics extendable to thewhole manifold, an Anosov torus. That is, T ⊂ M is an Anosov torus if thereexists a homeomorphism h : M →M such that h|T is homotopic to an Anosovdiffeomorphism. In [21] we obtained that the manifold must be again one of themanifolds of Conjecture 1.2 if it has an Anosov torus.

Theorem 1.7. A closed oriented irreducible 3-manifold admits an Anosov torusif and only if it is one of the following:

(1) the 3-torus(2) the mapping torus of −id

(3) the mapping torus of a hyperbolic automorphism

Let us recall that a 3-manifold is irreducible if any embedded 2-sphere boundsa ball. After the proof of the Poincare conjecture this is the same of havingtrivial second fundamental group. Three dimensional manifolds supporting apartially hyperbolic diffeomorphism are always irreducible thanks to Burago andIvanov results in [3]. Indeed, the existence of a Reebless foliation implies that themanifold is irreducible or it is S2 × S1.

Secondly suppose that there are no open accessibility classes. Then, accessibil-ity classes must foliate the whole manifold. Let us see that this foliation can nothave compact leaves. Observe that any such compact leaf must be a 2-torus. So,we have three possibilities: (1) there is an Anosov torus, (2) the set of compactleaves forms a strict non-trivial lamination, (3) the manifold is foliated by 2-tori.The first case has just been ruled out. In the second case, we would have that theboundary leaves contain a dense set of periodic points, as stated above, and hencethey would be Anosov tori again, which is impossible. Finally, in the third case,we conclude that the manifold is a fibration of tori over S1. This can only occur,in our setting, only if the manifold is the mapping torus of a diffeomorphismwhich commutes with an Anosov diffeomorphism as in Conjecture 1.2.

The following theorem is the first step in proving Conjectures 1.2 and 1.3. Seedefinitions in Subsection 2.1:

Theorem 1.8. Let f : M → M be a conservative partially hyperbolic diffeo-morphism of an orientable 3-manifold M . Suppose that the bundles Eσ are alsoorientable, σ = s, c, u, and that f is not accessible. Then one of the followingpossibilities holds:

(1) M is the mapping torus of a diffeomorphism which commutes with anAnosov diffeomorphism as in Conjecture 1.2.

(2) there is an f -invariant lamination ∅ 6= Γ(f) 6= M tangent to Es⊕Eu thattrivially extends to a (not necessarily invariant) foliation without compact


leaves of M . Moreover, the boundary leaves of Γ(f) are periodic, haveAnosov dynamics and dense periodic points.

(3) there is a minimal invariant foliation tangent to Es ⊕ Eu.

The assumption on the orientability of the bundles and M is not essential, infact, it can be achieved by a finite covering. The proof of Theorem 1.8 appearsat the end of Section 5.

We do not know of any example satisfying (2) in the theorem above. We havethe following question.

Question 1.9. Let f : N → N be an Anosov diffeomorphism on a completeRiemannian manifold N . Is it true that if Ω(f) = N then N is compact?

2. Preliminaries

2.1. Geometric preliminaries. In this section we state several definitions andconcepts that will be useful in the rest of this paper. From now on, M will be acompact connected Riemannian 3-manifold.

A lamination is a compact set Λ ⊂ M that can be covered by open charts U ⊂ Λwith a local product structure φ : U →Rn×T , where T is a locally compact subsetof Rk. On the overlaps Uα∩Uβ , the transition functions φβφ−1

α : Rn×T →Rn×T

are homeomorphisms and take the form:

φβ φ−1α (u, v) = (lαβ(u, v), tαβ(v)),

where lαβ are C1 with respect to the u variable. No differentiability is requiredin the transverse direction T . The sets φ−1(Rn × t) are called plaques. Eachpoint x of a lamination belongs to a maximal connected injectively immersedn-submanifold, called the leaf of x in L. The leaves are union of plaques. Observethat the leaves are C1, but vary only continuously. The number n is the dimension

of the lamination. If n = dim M − 1, we say Λ is a codimension-one lamination.The set L is an f -invariant lamination if it is a lamination such that f takes leavesinto leaves.

We call a lamination a foliation if Λ = M . In this case, we shall denote byF the set of leaves. In principle, we shall not assume any transverse differen-tiability. However, in case lαβ is Cr with respect to the v variable, we shall saythat the foliation is Cr. Note that even purely C0 codimension-one foliationsadmit a transverse 1-dimensional foliation (see Siebenmann [25], ). In our casethe existence of this 1-dimensional foliation is trivial thanks to the existence ofthe 1-dimensional center bundle Ec. These allows us to translate many localdeformation arguments, usually given in the C2 category, into the C0 categoryas observed, for instance, by Solodov [26]. In particular, Theorems 2.1 and 2.3,which were originally formulated for C2 foliations hold in the C0 case. We shallsay that a codimension-one foliation F , is transversely orientable if the transverse


1-dimensional foliation mentioned above is orientable. An invariant foliation is afoliation that is an invariant lamination.

Let Λ be a codimension-one lamination that is not a foliation. A complementary

region V is a component of M \ Λ. A closed complementary region V is themetric completion of a complementary region V with the path metric inducedby the Riemannian metric, the distance between two points being the infimumof the lengths of paths in V connecting them. A closed complementary region isindependent of the metric. Note that they are not necessarily compact. If Λ doesnot have compact leaves, then every closed complementary region decomposesinto a compact gut piece and non-compact interstitial regions which are I-bundlesover non-compact surfaces, and get thinner and thinner as they go away fromthe gut (see [13] or [8]). The interstitial regions meet the gut along annuli. Thedecomposition into interstitial regions and guts is unique up to isotopy. Moreover,one can take the interstitial regions as thin as one wishes.

A boundary leaf is a leaf corresponding to a component of ∂V , for V a closedcomplementary region. That is, a leaf is a non-boundary leaf if it is not containedin a closed complementary region.

Figure 1. A Reeb component

The geometry of codimension-one foliations is deeply related to the topology ofthe manifold that supports them. The following subset of a foliation is importantin their description. A Reeb component is a solid torus whose interior is foliatedby planes transverse to the of core of the solid torus, such that each leaf limitson the boundary torus, which is also a leaf (see Figure 1). A foliation that hasno Reeb components is called Reebless.

The following theorems show better the above mentioned relation:

Theorem 2.1 (Novikov). Let M be a compact orientable 3-manifold and F atransversely orientable codimension-one foliation. Then each of the followingimplies that F has a Reeb component:

(1) There is a closed, nullhomotopic transversal to F


(2) There is a leaf L in F such that π1(L) does not inject in π1(M)

The statement of this theorem can be found, for instance, in [6] Theorems 9.1.3.9.1.4., p.288. We shall also use the following theorem

Theorem 2.2 (Haefliger). Let Λ be a codimension one lamination in M . Thenthe set of points belonging to compact leaves is compact.

This theorem was originally formulated for foliations [10]. However, it alsoholds for laminations, see for instance [13].

We have the following consequence of Novikov’s Theorem about Reebless foli-ations. This theorem is stated in [24] as Corollary 2 on page 44.

Theorem 2.3. If M is a compact 3-manifold and F is a transversely orientablecodimension-one Reebless foliation, then either F is the product foliation of S2 ×S1, or F , the foliation induced by F on the universal cover M of M , is a foliationby planes R2. In particular, if M 6= S2 × S1 then M is irreducible.

This theorem was originally stated for C2 foliations, but it also holds for C0

foliations, due to Siebenmann’s theorem mentioned above.

2.2. Topologic preliminaries. Let M be a 3-dimensional manifold. A manifoldM is irreducible if every 2-sphere S2 embedded in the manifold bounds a 3-ball.Recall that a 2-torus T embedded in M is an Anosov torus if there exists adiffeomorphism f : M →M such that f(T ) = T and the action induced by f onπ1(T ), that is, f#|T : π1(T )→π1(T ), is a hyperbolic automorphism. Equivalently,f restricted to T is isotopic to a hyperbolic automorphism.

We shall assume from now on, that M is an irreducible 3-manifold since thisis the case for 3-manifolds supporting partially hyperbolic diffeomorphisms. Inthis subsection, we will focus on what is called the JSJ-decomposition of M (seebelow). That is, we will cut M along certain kind of tori, called incompressible,and will obtain certain 3-manifolds with boundary that are easier to handle, whichare, respectively, Seifert manifolds, and atoroidal and acylindrical manifolds. Letus introduce these definitions first.

An orientable surface S embedded in M is incompressible if the homomorphisminduced by the inclusion map i# : π1(S) → π1(M) is injective; or, equivalently,if there is no embedded disc D2 ⊂ M such that D ∩ S = ∂D and ∂D ≁ 0 in S

(see, for instance, [12, Page 10]). We also require that S 6= S2.A manifold with or without boundary is Seifert, if it admits a one dimensional

foliation by closed curves, called a Seifert fibration. The boundary of an orientableSeifert manifold with boundary consists of finite union of tori. There are manyexamples of Seifert manifolds, for instance S3, T1S where S is a surface, etc.

The other type of manifold obtained in the JSJ-decomposition is atoroidaland acylindrical manifolds. A 3-manifold with boundary N is atoroidal if every


incompressible torus is ∂-parallel, that is, isotopic to a subsurface of ∂N . A 3-manifold with boundary N is acylindrical if every incompressible annulus A thatis properly embedded, i.e. ∂A ⊂ ∂N , is ∂-parallel, by an isotopy fixing ∂A.

As we mentioned before, a closed irreducible 3-manifold admits a natural de-composition into Seifert pieces on one side, and atoroidal and acylindrical com-ponents on the other:

Theorem 2.4 (JSJ-decomposition [14], [15]). If M is an irreducible closed ori-entable 3-manifold, then there exists a collection of disjoint incompressible tori Tsuch that each component of M \T is either Seifert, or atoroidal and acylindrical.Any minimal such collection is unique up to isotopy. This means, if T is a col-lection as described above, it contains a minimal sub-collection m(T ) satisfyingthe same claim. All collections m(T ) are isotopic.

Any minimal family of incompressible tori as described above is called a JSJ-decomposition of M . When it is clear from the context we shall also call JSJ-decomposition the set of pieces obtained by cutting the manifold along these tori.Note that if M is either atoroidal or Seifert, then T = ∅.

2.3. Dynamic preliminaries. Throughout this paper we shall work with a par-

tially hyperbolic diffeomorphism f , that is, a diffeomorphism admitting a non-trivialTf -invariant splitting of the tangent bundle TM = Es ⊕ Ec ⊕ Eu, such that allunit vectors vσ ∈ Eσ

x (σ = s, c, u) with x ∈ M verify:

‖Txfvs‖ < ‖Txfvc‖ < ‖Txfvu‖

for some suitable Riemannian metric. f also must satisfy that ‖Tf |Es‖ < 1 and‖Tf−1|Eu‖ < 1. We shall say that a partially hyperbolic diffeomorphism f thatsatisfies

‖Txfvs‖ < ‖Tyfvc‖ < ‖Tzfvu‖ ∀x, y, z ∈ M

is absolutely partially hyperbolic.

We shall also assume that f is conservative, i.e. it preserves Lebesgue measureassociated to a smooth volume form.

It is a known fact that there are foliations Wσ tangent to the distributions Eσ

for σ = s, u (see for instance [2]). The leaf of Wσ containing x will be calledW σ(x), for σ = s, u. The connected component of x in the intersection of W s(x)with a small ε-ball centered at x is the ε-local stable manifold of x, and is denotedby W s

ε (x).In general it is not true that there is a foliation tangent to Ec. It is false even

in case dim Ec = 1 (see [22]). However, in Proposition 3.4 of [1] it is shown thatif dim Ec = 1, then f is weakly dynamically coherent. This means, for each x ∈ M

there are complete immersed C1 manifolds which contain x and are everywheretangent to Ec, Ecs and Ecu, respectively. We will call a center curve any curvewhich is everywhere tangent to Ec. Moreover, we will use the following fact:


Proposition 2.5 ([1]). If γ is a center curve through x, then

W sε (γ) =

⋃

y∈γ

W sε (y) and W u

ε (γ) =⋃

y∈γ

W uε (y)

are C1 immersed manifolds everywhere tangent to Es ⊕ Ec and Ec ⊕ Eu respec-tively.

We shall say that a set X is s-saturated or u-saturated if it is a union of leaves ofthe strong foliations Ws or Wu respectively. We also say that X is su-saturatedif it is both s- and u-saturated. The accessibility class AC(x) of the point x ∈ M

is the minimal su-saturated set containing x. Note that the accessibility classesform a partition of M . If there is some x ∈ M whose accessibility class is M , thenthe diffeomorphism f is said to have the accessibility property. This is equivalentto say that any two points of M can be joined by a path which is piecewisetangent to Es or to Eu. A diffeomorphism is said to be essentially accessible ifany su-saturated set has full or null measure.

The theorem below relates accessibility with ergodicity. In fact it is proven ina more general setting, but we shall use the following formulation:

Theorem 2.6 ([4],[18]). If f is a C2 conservative partially hyperbolic diffeomor-phism with the (essential) accessibility property and dim Ec = 1, then f is ergodic.

In [20] it is proved that there are manifolds whose topology implies the ac-cessibility property holds for all partially hyperbolic diffeomorphisms. In thesemanifolds, all partially hyperbolic diffeomorphisms are ergodic.

Sometimes we will focus on the openness of the accessibility classes. Note thatthe accessibility classes form a partition of M . If all of them are open then, infact, f has the accessibility property. We will call U(f) = x ∈ M ; AC(x) isopen and Γ(f) = M \ U(f). Note that f has the accessibility property if andonly if Γ(f) = ∅. We have the following property of non-open accessibility classes:

Proposition 2.7 ([18]). The set Γ(f) is a codimension-one lamination, havingthe accessibility classes as leaves.

In fact, any compact su-saturated subset of Γ(f) is a lamination.

The above proposition is Proposition A.3. of [18]. The fact that the leaves ofΓ(f) are C1 may be found in [7]. The following proposition is Proposition A.5 of[18]:

Proposition 2.8 ([18]). If Λ is an invariant sub-lamination of Γ(f), then eachboundary leaf of Λ is periodic and the periodic points are dense in it (with theinduced topology).

Moreover, the stable and unstable manifolds of each periodic point are dense ineach plaque of a boundary leaf of Λ.


Observe that the proof of Proposition A.5 of [18] shows in fact that periodicpoints are dense in the accessibility classes of the boundary leaves of V endowedwith its intrinsic topology. In other words, periodic points are dense in eachplaque of the boundary leaves of V .

We shall also use the following theorem by Brin, Burago and Ivanov, whoseproof is in [1], after Proposition 2.1.

Theorem 2.9 ([1]). If f : M3 →M3 is a partially hyperbolic diffeomorphism,and there is an open set V foliated by center-unstable leaves, then there cannotbe a closed center-unstable leaf bounding a solid torus in V .

3. Anosov tori

In this section we will say a few words about the proof of Theorem 1.7. Theidea in its proof is that, given an Anosov torus T , we can “place” T so that eitherT belongs to the family T given by the JSJ-decomposition (Theorem 2.4), or elseT is in a Seifert component, and it is either transverse to all fibers, or it is unionof fibers of this Seifert component. See Proposition 3.3.

It is important to note the following property of Anosov tori:

Theorem 3.1 ([20]). Anosov tori are incompressible.

An Anosov torus in an atoroidal component will then be ∂-parallel to a compo-nent of its boundary. In this case, we can assume T ∈ T . On the other hand, theTheorem of Waldhausen below, guarantees that we can always place an incom-pressible torus in a Seifert manifold in a “standard” form; namely, the following:a surface is horizontal in a Seifert manifold if it is transverse to all fibers, andvertical if it is union of fibers:

Theorem 3.2 (Waldhausen [27]). Let M be a compact connected Seifert manifold,with or without boundary. Then any incompressible surface can be isotoped to behorizontal or vertical.

The architecture of the proof of Theorem 1.7 is contained in the followingproposition.

Proposition 3.3. Let T be an Anosov torus of a closed irreducible orientablemanifold M . Then, there exists a diffeomorphism f : M → M and a JSJ-decomposition T such that

(1) f |T is a hyperbolic toral automorphism,(2) f(T ) = T , and(3) one of the following holds

(a) T ∈ T(b) T is a vertical torus in a Seifert component of M \ T , and T is not

∂-parallel in this component.


(c) M is a Seifert manifold (T = ∅), and T is a horizontal torus,

The proposition above allows us to split the proof of Theorem 1.7 into cases.Note that case (3b) includes the case in which M is a Seifert manifold and T isa vertical torus.

In the case that T is a vertical torus in a Seifert component we can cut thiscomponent along T . Then we can suppose that T is in the boundary. We takeprofit of the fact that in most manifolds the Seifert fibration is unique up toisotopy. Since the dynamics restricted to T is Anosov we have that the manifoldhas more than one Seifert fibration. This lead us to show that this Seifert com-ponent must be T2 × [0, 1]. This gives that the whole manifold must be one ofthe manifolds of Theorem 1.7.

If T is horizontal torus then the manifold M is Seifert and T intersects all thefibers. This is discarded in a case by case study thanks to the fact that the Seifertmanifolds having horizontal torus a finite.

The last and more difficult case is when T is part of the JSJ-decompositionbut it is not the boundary of a Seifert component. The proof in this case iscomplicated but a very rough idea is to take a properly embedded surface S

with an essential circle of T in its boundary. Taking a large iterate fn(S) andconsidering S ∩ fn(S), it is possible to construct a non-parallel incompressiblecylinder as a union of a band in S and a band in fn(S). This leads to contradictionbecause the component is not Seifert and then, it is acylindrical.

4. The su-lamination Γ(f)

Let f be a partially hyperbolic diffeomorphism of a compact 3-manifold M .From Subsection 2.3 it follows that we have three possibilities: (1) f has theaccessibility property, (2) the set of non-open accessibility classes is a strict lam-ination, ∅ Γ(f) M or (3) the set of non-open accessibility classes foliates M :Γ(f) = M .

Now, we shall distinguish two possible cases in situations (2) and (3):

(a) the lamination Γ(f) does not contain compact leaves(b) the lamination Γ(f) contains compact leaves

In this section we deal with the case (2a). In fact, for our purposes it will besufficient to assume that there exists an f -invariant sub-lamination Λ of Γ(f)without compact leaves. Section 5 treats the cases (2b) and (3b). Section 6treats the case (3a).

In this section, we will prove that the complement of Λ consists of I-bundles.To this end, we shall assume that the bundles Eσ (σ = s, c, u) and the manifoldM are orientable (we can achieve this by considering a finite covering).


Theorem 4.1 ([20], Theorem 4.1). If ∅ Λ ⊂ Γ(f) is an orientable and trans-versely orientable f -invariant sub-lamination without compact leaves such thatΛ 6= M , then all closed complementary regions of Λ are I-bundles.

Theorem 4.1 was proved by showing:

Proposition 4.2. Let Λ ⊂ Γ(f) be a nonempty f -invariant sub-lamination with-out compact leaves. Then Ec is uniquely integrable in the closed complementaryregions of Λ.

The proof of this proposition is rather technical. The interested reader mayfound a proof in [20].

Let us consider V a closed complementary region of Λ, and call I(V ) the union

of all interstitial regions of V and G(V ) the gut of V (see Subsection 2.1), so that

V = I(V ) ∪ G(V ).

The following statement is rather standard:

Lemma 4.3. Let f : M →M be a partially hyperbolic diffeomorphism. If U is anopen invariant set such that U ⊂ Ω(f), then the closure of U is su-saturated.

Let us observe that if V is connected then there are only two boundary leavesof V . Indeed, as we mentioned before periodic points are dense in boundaryleaves. This fact jointly with the local product structure imply, using standardarguments, that the stable and unstable leaves of periodic points are dense too.Take a periodic point p in a boundary leaf and in the intersticial region. Thereare center curves joining the points in the local stable manifold of p with otherboundary curve L1 of V (the same property holds for the local unstable manifold).Invariance of the stable manifold of p and boundary leaves give that the centercurve of any point of the stable manifold joins the boundary leaf L0 containingp with L1. Denseness of the stable and unstable manifolds of p implies that thecomplement of the set of points such that their center manifold join L0 with L1

is totally disconnected. Then, it is not difficult to see that L0 and L1 are theunique boundary leaves of V .

Also, since periodic points are dense in the boundary leaves due to Proposition2.8, there is an iterate of f that fixes all connected components of V , so we willassume when proving Theorem 4.1 that V is connected and has two boundaryleaves L0 and L1.

Proof of Theorem 4.1. We will present a sketch of a different approach to a proofthan the one given in [20]. The strategy will be to show that all center leaves in

V meet both L0 and L1. Let p be a periodic point in L0∩I(V ). As we mentionedbefore its center leaf meets L1, and the same happens for all points in its stableand unstable manifolds. Now stable and unstable manifolds of a periodic point


are dense in each plaque of L0 (Proposition 2.8). So the set of points in L0 whosecenter leaf does not reach L1 is contained in a totally disconnected set.

Let us suppose that x0 is a point in L0 whose center leaf does not reach L1.Then, since center curves of points of the intersticial region clearly reach theboundary, W c(x0) is contained in G(V ). Take a small rectangle R in L0 aroundx0 formed by arcs of stable and unstable manifolds of a periodic point. Moreover,we can assume that the center curves of the points of R0 reach L1. Of course,the image is another rectangle R1 formed by stable and unstable arcs. Then,the center arcs of the points of R0 and the interiors of R0 and R1 form a 2-sphere S. Since Rosenberg’s theorem [23] remains valid in this setting and V is

foliated by Wcs that is Reebless and transverse to the boundary, we have that V

is irreducible. Then, S bounds a ball B. Now, since W c(x0) does not reach L1

and is contained in B, it accumulates in B but Novikov’s Theorem implies theexistence of a Reeb component, a contradiction.

Theorem 4.1 implies that any non trivial invariant sub-lamination Λ ⊂ Γ(f)without compact leaves can be extended to a foliation of M without compactleaves. Indeed, any complementary region V is an I-bundle, and hence it isdiffeomorphic to the product of a boundary leaf times the open interval: L0 ×(0, 1). The foliation Ft = L0 × t induces a foliation of V .

This has the following consequence in case the fundamental group of M isnilpotent:

Proposition 4.4. If M is a compact 3-manifold with nilpotent fundamentalgroup, and ∅ ( Λ ( M , is an invariant sub-lamination of Γ(f), then thereexists a leaf of Λ that is a periodic 2-torus with Anosov dynamics.

Proof. If Λ has a compact leaf, let us consider the set Λc of all compact leavesof Λ. Λc is in fact an invariant sub-lamination, due to Theorem 2.2. HenceProposition 2.8 implies that the boundary leaves of Λc are periodic 2-tori withAnosov dynamics, and we obtain the claim.

If, on the contrary, Λ does not have compact leaves, then due to Theorem4.1 above, we can extend Λ to a foliation F of M without compact leaves. Inparticular, F is a Reebless foliation. Item (2) of Theorem 2.1 implies that for allboundary leaves L of Λ, π1(L) injects in π1(M), and is therefore nilpotent.

Now, this implies that the boundary leaves can only be planes or cylinders.Theorem 2.8 implies that stable and unstable leaves of periodic points are densein those leaves, which is impossible for the case of the plane or the cylinder.Therefore, Λ must contain a compact leaf, and due to what was shown above, itmust contain a periodic 2-torus with Anosov dynamics.

In fact, Theorem 1.7 implies that periodic 2-tori with Anosov dynamics are notpossible in 3-manifolds with nilpotent fundamental group, unless the manifold is


T3. Hence the hypotheses of Proposition 4.4 are not fulfilled, unless the manifoldis T3. This will eliminate case (2) mentioned at the beginning of this section.

5. The trichotomy of Theorem 1.8

In this section we will prove Theorem 1.8. This theorem and the results in thissection are valid for any 3-manifold M , and do not require that its fundamentalgroup be nilpotent. Moreover, Theorem 3.1 does not even require the existenceof a partially hyperbolic diffeomorphism.

Let T be an embedded 2-torus in M . We shall call T an Anosov torus ifthere exists a homeomorphism g : M →M such that T is g-invariant, and g|T ishomotopic to an Anosov diffeomorphism.

Also, let S be a two-sided embedded closed surface of M3 other than the sphere.S is incompressible if and only if the homomorphism induced by the inclusion mapi# : π1(S) → π1(M) is injective; or, equivalently, after the Loop Theorem, if thereis no embedded disc D2 ⊂ M such that D ∩ S = ∂D and ∂D ≁ 0 in S (see, forinstance, [12]).

Recall that Theorem 3.1 says that Anosov tori are incompressible. We insistthat this theorem is general, and does not depend on the existence of a partiallyhyperbolic dynamics in the manifold.

We also need the following fact about codimension one laminations.

Theorem 5.1. Let F be a codimension one C0-foliation without compact leavesof a three dimensional compact manifold M . Then, F has a finite number ofminimal sets.

We are now in position to prove Theorem 1.8 of Page 4:

Proof of Theorem 1.8. If Γ(f) = M then there are no Reeb components. Indeed,since f is conservative, if there were a Reeb component, then its boundary torusshould be periodic. We get a contradiction from Theorem ??. This gives case (3)except the minimality.

Let us assume that Γ(f) 6= M . If Γ(f) contains a compact leaf then the set ofcompact leaves is a sub-lamination Λ of Γ(f) by Theorem 2.2. Proposition 2.8implies that the boundary leaves of Λ are Anosov tori, and we obtain case (1) asa consequence of Theorem 1.7.

If Γ(f) 6= M and contains no compact leaves, then Theorem 4.1 and Proposi-tion 2.8 give us case (2).

Finally we show minimality in case (3). On the one hand, if Γ(f) = M andhas a compact leaf we have two possibilities: either all leaves are compact ornot. If not then, the previous argument implies the existence of an Anosov torusand we are in case (1). If all leaves are compact, as we mentioned before, themanifold is a torus bundle and the hyperbolic dynamics on fibers implies thatwe are again in case (1). On the other hand, if Γ(f) has no compact leaves and


has a minimal sub-lamination L, we have that L is periodic (recall that minimalsub-laminations of a codimension one foliation are finite, Theorem 5.1). Then,we are again in case (2).

6. Nilmanifolds

Let f : M →M be a conservative partially hyperbolic diffeomorphism of acompact orientable three dimensional nilmanifold M 6= T3. As consequence ofProposition 4.4 and Theorems 1.7 and 1.7 we have that Es ⊕ Eu integrates to aminimal foliation F su if f does not have the accessibility property. Indeed theonly possibilities in the trichotomy of Theorem 1.8 are (2) and (3) and Proposi-tion 4.4 says that there is an Anosov torus if we are in case (2). But this lastcase is impossible for a nilmanifold M 6= T3. In this section we shall give somearguments showing that the existence of such a foliation leads us to a contradic-tion. In [20] the reader can find a different proof of the same fact. Without lossof generality we may assume, by taking a double covering if necessary, that F su

is transversely orientable. Observe that the double covering of a nilmanifold isagain a nilmanifold.

The first step is that Parwani [16] proved (following Burago-Ivanov [3] argu-ments) that the action induced by f in the first homology group of M is hyper-bolic. By duality the same is true for the first cohomology group.

The second step is given by Plante results in [17] (see also [13]). Since F su isa minimal foliation of a manifold whose fundamental group has non-exponentialgrowth there exists a transverse holonomy invariant measure µ of full support.This measure is unique up to multiplication by a constant and represents anelement of the first cohomology group of M . The action of f leaves F su invariantand induces a new transverse measure ν, an image of former one. The uniquenessimplies that ν = λµ for some λ > 0. Since of the action of f on H1(M) ishyperbolic, then λ 6= 1. Suppose that λ > 1 (if the contrary is true take f−1).

The third step is to observe that λ > 1 implies that f is expanding the µ mea-sure of center curves. Since µ has full support and the su−bundle is hyperbolicwe would obtain that f is conjugated to Anosov leading to contradiction withthe fact that M 6= T3.

7. M = T3

In this section we present the results announced by Hammerlindl and Ures onConjecture 1.3, that the nonexistence of nonergodic partially hyperbolic diffeo-morphisms homotopic to Anosov in dimension 3. They are able to prove thefollowing result.

Theorem 7.1 ([11]). Let f : T3 → T3 be a C1+α conservative partially hyperbolicdiffeomorphism homotopic to a hyperbolic automorphism A. Suppose that f isnot ergodic. Then,


(1) Es × Eu integrates to a minimal foliation.(2) f is topologically conjugated to A and the conjugacy sends strong leaves

of f into the corresponding strong leaves of A.(3) The center Lyapunov exponent is 0 a.e.

We remark that it is not known if there exists a diffeomorphism satisfying theconditions of the theorem above.

Now, in order to prove Conjecture 1.3 we have two possibilities: either weprove that a diffeomorphism satisfying the conditions of Theorem 7.1 is ergodicor we prove that such a diffeomorphism cannot exist. Hammerlindl and Uresannounced that if f is C2 and the center stable and center unstable leaves of aperiodic point are C2 then, f is ergodic.

References

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[2] M. Brin, Ya Pesin, Partially hyperbolic dynamical systems, Math. USSR Izv.8, (1974)177-218.

[3] D. Burago, S. Ivanov, Partially hyperbolic diffeomorphisms of 3-manifolds with abelianfundamental groups, preprint (2007)

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ideo, Uruguay

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ACCESSIBILITY AND ABUNDANCE OF ERGODICITY IN DIMENSION THREE: A SURVEY